Problem: Simplify the following expression and state the condition under which the simplification is valid: $y = \dfrac{z^2 + 5z + 4}{z^2 + 10z + 24}$
First factor the expressions in the numerator and denominator. $ \dfrac{z^2 + 5z + 4}{z^2 + 10z + 24} = \dfrac{(z + 1)(z + 4)}{(z + 6)(z + 4)} $ Notice that the term $(z + 4)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z + 4)$ gives: $y = \dfrac{z + 1}{z + 6}$ Since we divided by $(z + 4)$, $z \neq -4$. $y = \dfrac{z + 1}{z + 6}; \space z \neq -4$